# Properties

 Label 1568.2.i Level $1568$ Weight $2$ Character orbit 1568.i Rep. character $\chi_{1568}(961,\cdot)$ Character field $\Q(\zeta_{3})$ Dimension $80$ Newform subspaces $25$ Sturm bound $448$ Trace bound $25$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$1568 = 2^{5} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1568.i (of order $$3$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$7$$ Character field: $$\Q(\zeta_{3})$$ Newform subspaces: $$25$$ Sturm bound: $$448$$ Trace bound: $$25$$ Distinguishing $$T_p$$: $$3$$, $$5$$, $$11$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(1568, [\chi])$$.

Total New Old
Modular forms 512 80 432
Cusp forms 384 80 304
Eisenstein series 128 0 128

## Trace form

 $$80 q - 40 q^{9} + O(q^{10})$$ $$80 q - 40 q^{9} + 16 q^{13} - 32 q^{25} - 16 q^{29} + 8 q^{33} - 8 q^{37} - 16 q^{41} + 8 q^{45} + 8 q^{53} + 80 q^{57} + 24 q^{61} - 24 q^{65} - 24 q^{73} - 96 q^{81} + 112 q^{85} - 40 q^{89} - 24 q^{93} + 16 q^{97} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(1568, [\chi])$$ into newform subspaces

Label Dim $A$ Field CM Traces $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
1568.2.i.a $2$ $12.521$ $$\Q(\sqrt{-3})$$ None $$0$$ $$-2$$ $$-2$$ $$0$$ $$q+(-2+2\zeta_{6})q^{3}-2\zeta_{6}q^{5}-\zeta_{6}q^{9}+\cdots$$
1568.2.i.b $2$ $12.521$ $$\Q(\sqrt{-3})$$ None $$0$$ $$-2$$ $$0$$ $$0$$ $$q+(-2+2\zeta_{6})q^{3}-\zeta_{6}q^{9}+(-4+4\zeta_{6})q^{11}+\cdots$$
1568.2.i.c $2$ $12.521$ $$\Q(\sqrt{-3})$$ None $$0$$ $$-2$$ $$0$$ $$0$$ $$q+(-2+2\zeta_{6})q^{3}-\zeta_{6}q^{9}+(4-4\zeta_{6})q^{11}+\cdots$$
1568.2.i.d $2$ $12.521$ $$\Q(\sqrt{-3})$$ None $$0$$ $$-2$$ $$2$$ $$0$$ $$q+(-2+2\zeta_{6})q^{3}+2\zeta_{6}q^{5}-\zeta_{6}q^{9}+\cdots$$
1568.2.i.e $2$ $12.521$ $$\Q(\sqrt{-3})$$ $$\Q(\sqrt{-1})$$ $$0$$ $$0$$ $$-4$$ $$0$$ $$q-4\zeta_{6}q^{5}+3\zeta_{6}q^{9}-4q^{13}+(-8+\cdots)q^{17}+\cdots$$
1568.2.i.f $2$ $12.521$ $$\Q(\sqrt{-3})$$ $$\Q(\sqrt{-1})$$ $$0$$ $$0$$ $$-2$$ $$0$$ $$q-2\zeta_{6}q^{5}+3\zeta_{6}q^{9}-6q^{13}+(2-2\zeta_{6})q^{17}+\cdots$$
1568.2.i.g $2$ $12.521$ $$\Q(\sqrt{-3})$$ $$\Q(\sqrt{-1})$$ $$0$$ $$0$$ $$2$$ $$0$$ $$q+2\zeta_{6}q^{5}+3\zeta_{6}q^{9}+6q^{13}+(-2+\cdots)q^{17}+\cdots$$
1568.2.i.h $2$ $12.521$ $$\Q(\sqrt{-3})$$ $$\Q(\sqrt{-1})$$ $$0$$ $$0$$ $$4$$ $$0$$ $$q+4\zeta_{6}q^{5}+3\zeta_{6}q^{9}+4q^{13}+(8-8\zeta_{6})q^{17}+\cdots$$
1568.2.i.i $2$ $12.521$ $$\Q(\sqrt{-3})$$ None $$0$$ $$2$$ $$-2$$ $$0$$ $$q+(2-2\zeta_{6})q^{3}-2\zeta_{6}q^{5}-\zeta_{6}q^{9}+(4+\cdots)q^{11}+\cdots$$
1568.2.i.j $2$ $12.521$ $$\Q(\sqrt{-3})$$ None $$0$$ $$2$$ $$0$$ $$0$$ $$q+(2-2\zeta_{6})q^{3}-\zeta_{6}q^{9}+(-4+4\zeta_{6})q^{11}+\cdots$$
1568.2.i.k $2$ $12.521$ $$\Q(\sqrt{-3})$$ None $$0$$ $$2$$ $$0$$ $$0$$ $$q+(2-2\zeta_{6})q^{3}-\zeta_{6}q^{9}+(4-4\zeta_{6})q^{11}+\cdots$$
1568.2.i.l $2$ $12.521$ $$\Q(\sqrt{-3})$$ None $$0$$ $$2$$ $$2$$ $$0$$ $$q+(2-2\zeta_{6})q^{3}+2\zeta_{6}q^{5}-\zeta_{6}q^{9}+(-4+\cdots)q^{11}+\cdots$$
1568.2.i.m $4$ $12.521$ $$\Q(\sqrt{-3}, \sqrt{5})$$ None $$0$$ $$-2$$ $$-2$$ $$0$$ $$q+(-1-\beta _{1}-\beta _{2})q^{3}+(\beta _{1}-\beta _{2}-\beta _{3})q^{5}+\cdots$$
1568.2.i.n $4$ $12.521$ $$\Q(\sqrt{-3}, \sqrt{5})$$ None $$0$$ $$-2$$ $$2$$ $$0$$ $$q+(-1-\beta _{1}-\beta _{2})q^{3}+(-\beta _{1}+\beta _{2}+\cdots)q^{5}+\cdots$$
1568.2.i.o $4$ $12.521$ $$\Q(\sqrt{2}, \sqrt{-3})$$ None $$0$$ $$-2$$ $$2$$ $$0$$ $$q+(-1+\beta _{1}-\beta _{2})q^{3}+(2\beta _{1}-\beta _{2}+2\beta _{3})q^{5}+\cdots$$
1568.2.i.p $4$ $12.521$ $$\Q(\sqrt{-3}, \sqrt{7})$$ None $$0$$ $$0$$ $$-6$$ $$0$$ $$q+\beta _{1}q^{3}+3\beta _{2}q^{5}+4\beta _{2}q^{9}+\beta _{1}q^{11}+\cdots$$
1568.2.i.q $4$ $12.521$ $$\Q(\sqrt{2}, \sqrt{-3})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{1}q^{3}-\beta _{2}q^{9}+(-2-2\beta _{2})q^{11}+\cdots$$
1568.2.i.r $4$ $12.521$ $$\Q(\sqrt{2}, \sqrt{-3})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{1}q^{3}-\beta _{2}q^{9}+(2+2\beta _{2})q^{11}-2\beta _{3}q^{13}+\cdots$$
1568.2.i.s $4$ $12.521$ $$\Q(\sqrt{2}, \sqrt{-3})$$ $$\Q(\sqrt{-1})$$ $$0$$ $$0$$ $$0$$ $$0$$ $$q+3\beta _{1}q^{5}+(3+3\beta _{2})q^{9}+5\beta _{3}q^{13}+\cdots$$
1568.2.i.t $4$ $12.521$ $$\Q(\sqrt{2}, \sqrt{-3})$$ $$\Q(\sqrt{-1})$$ $$0$$ $$0$$ $$0$$ $$0$$ $$q-\beta _{1}q^{5}+(3+3\beta _{2})q^{9}+\beta _{3}q^{13}+(-5\beta _{1}+\cdots)q^{17}+\cdots$$
1568.2.i.u $4$ $12.521$ $$\Q(\zeta_{12})$$ None $$0$$ $$0$$ $$2$$ $$0$$ $$q-\zeta_{12}^{2}q^{3}+(1-\zeta_{12})q^{5}+3\zeta_{12}^{2}q^{11}+\cdots$$
1568.2.i.v $4$ $12.521$ $$\Q(\sqrt{-3}, \sqrt{5})$$ None $$0$$ $$2$$ $$-2$$ $$0$$ $$q+(1+\beta _{1}+\beta _{2})q^{3}+(\beta _{1}-\beta _{2}-\beta _{3})q^{5}+\cdots$$
1568.2.i.w $4$ $12.521$ $$\Q(\sqrt{-3}, \sqrt{5})$$ None $$0$$ $$2$$ $$2$$ $$0$$ $$q+(1+\beta _{1}+\beta _{2})q^{3}+(-\beta _{1}+\beta _{2}+\beta _{3})q^{5}+\cdots$$
1568.2.i.x $4$ $12.521$ $$\Q(\sqrt{2}, \sqrt{-3})$$ None $$0$$ $$2$$ $$2$$ $$0$$ $$q+(1+\beta _{1}+\beta _{2})q^{3}+(-2\beta _{1}-\beta _{2}-2\beta _{3})q^{5}+\cdots$$
1568.2.i.y $8$ $12.521$ 8.0.207360000.1 None $$0$$ $$0$$ $$0$$ $$0$$ $$q-\beta _{2}q^{3}+(-2\beta _{1}+2\beta _{4})q^{5}-7\beta _{3}q^{9}+\cdots$$

## Decomposition of $$S_{2}^{\mathrm{old}}(1568, [\chi])$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(1568, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(28, [\chi])$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(49, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(56, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(98, [\chi])$$$$^{\oplus 5}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(112, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(196, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(224, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(392, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(784, [\chi])$$$$^{\oplus 2}$$